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Theorem ltexpri 7816

Description: Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)

**Assertion**
Ref	Expression
ltexpri	⊢ (𝐴<_P 𝐵 → ∃𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

Proof of Theorem ltexpri Dummy variables 𝑦 𝑧 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . . . 8 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
2	1	eleq1d 2298	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝑧 ∈ (2^nd ‘𝐴) ↔ 𝑣 ∈ (2^nd ‘𝐴)))
3		simpl 109	. . . . . . . . 9 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → 𝑦 = 𝑢)
4	1, 3	oveq12d 6028	. . . . . . . 8 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝑧 +_Q 𝑦) = (𝑣 +_Q 𝑢))
5	4	eleq1d 2298	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵) ↔ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵)))
6	2, 5	anbi12d 473	. . . . . 6 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵)) ↔ (𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))))
7	6	cbvexdva 1976	. . . . 5 ⊢ (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵)) ↔ ∃𝑣(𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))))
8	7	cbvrabv 2798	. . . 4 ⊢ {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))} = {𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))}
9	1	eleq1d 2298	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝑧 ∈ (1^st ‘𝐴) ↔ 𝑣 ∈ (1^st ‘𝐴)))
10	4	eleq1d 2298	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵) ↔ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵)))
11	9, 10	anbi12d 473	. . . . . 6 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵)) ↔ (𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))))
12	11	cbvexdva 1976	. . . . 5 ⊢ (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵)) ↔ ∃𝑣(𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))))
13	12	cbvrabv 2798	. . . 4 ⊢ {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))} = {𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))}
14	8, 13	opeq12i 3862	. . 3 ⊢ ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ = ⟨{𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))}, {𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))}⟩
15	14	ltexprlempr 7811	. 2 ⊢ (𝐴<_P 𝐵 → ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ ∈ P)
16	14	ltexprlemfl 7812	. . . 4 ⊢ (𝐴<_P 𝐵 → (1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) ⊆ (1^st ‘𝐵))
17	14	ltexprlemrl 7813	. . . 4 ⊢ (𝐴<_P 𝐵 → (1^st ‘𝐵) ⊆ (1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)))
18	16, 17	eqssd 3241	. . 3 ⊢ (𝐴<_P 𝐵 → (1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (1^st ‘𝐵))
19	14	ltexprlemfu 7814	. . . 4 ⊢ (𝐴<_P 𝐵 → (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) ⊆ (2^nd ‘𝐵))
20	14	ltexprlemru 7815	. . . 4 ⊢ (𝐴<_P 𝐵 → (2^nd ‘𝐵) ⊆ (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)))
21	19, 20	eqssd 3241	. . 3 ⊢ (𝐴<_P 𝐵 → (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (2^nd ‘𝐵))
22		ltrelpr 7708	. . . . . . 7 ⊢ <_P ⊆ (P × P)
23	22	brel 4773	. . . . . 6 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
24	23	simpld 112	. . . . 5 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
25		addclpr 7740	. . . . 5 ⊢ ((𝐴 ∈ P ∧ ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ ∈ P) → (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) ∈ P)
26	24, 15, 25	syl2anc 411	. . . 4 ⊢ (𝐴<_P 𝐵 → (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) ∈ P)
27	23	simprd 114	. . . 4 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
28		preqlu 7675	. . . 4 ⊢ (((𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵 ↔ ((1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (1^st ‘𝐵) ∧ (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (2^nd ‘𝐵))))
29	26, 27, 28	syl2anc 411	. . 3 ⊢ (𝐴<_P 𝐵 → ((𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵 ↔ ((1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (1^st ‘𝐵) ∧ (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (2^nd ‘𝐵))))
30	18, 21, 29	mpbir2and 950	. 2 ⊢ (𝐴<_P 𝐵 → (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵)
31		oveq2 6018	. . . 4 ⊢ (𝑥 = ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ → (𝐴 +_P 𝑥) = (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩))
32	31	eqeq1d 2238	. . 3 ⊢ (𝑥 = ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ → ((𝐴 +_P 𝑥) = 𝐵 ↔ (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵))
33	32	rspcev 2907	. 2 ⊢ ((⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ ∈ P ∧ (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵) → ∃𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)
34	15, 30, 33	syl2anc 411	1 ⊢ (𝐴<_P 𝐵 → ∃𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1395 ∃wex 1538 ∈ wcel 2200 ∃wrex 2509 {crab 2512 ⟨cop 3669 class class class wbr 4083 ‘cfv 5321 (class class class)co 6010 1^st c1st 6293 2^nd c2nd 6294 Qcnq 7483 +_Q cplq 7485 Pcnp 7494 +_P cpp 7496 <_P cltp 7498

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4259 ax-pr 4294 ax-un 4525 ax-setind 4630 ax-iinf 4681

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-eprel 4381 df-id 4385 df-po 4388 df-iso 4389 df-iord 4458 df-on 4460 df-suc 4463 df-iom 4684 df-xp 4726 df-rel 4727 df-cnv 4728 df-co 4729 df-dm 4730 df-rn 4731 df-res 4732 df-ima 4733 df-iota 5281 df-fun 5323 df-fn 5324 df-f 5325 df-f1 5326 df-fo 5327 df-f1o 5328 df-fv 5329 df-ov 6013 df-oprab 6014 df-mpo 6015 df-1st 6295 df-2nd 6296 df-recs 6462 df-irdg 6527 df-1o 6573 df-2o 6574 df-oadd 6577 df-omul 6578 df-er 6693 df-ec 6695 df-qs 6699 df-ni 7507 df-pli 7508 df-mi 7509 df-lti 7510 df-plpq 7547 df-mpq 7548 df-enq 7550 df-nqqs 7551 df-plqqs 7552 df-mqqs 7553 df-1nqqs 7554 df-rq 7555 df-ltnqqs 7556 df-enq0 7627 df-nq0 7628 df-0nq0 7629 df-plq0 7630 df-mq0 7631 df-inp 7669 df-iplp 7671 df-iltp 7673

This theorem is referenced by: lteupri 7820 ltaprlem 7821 ltaprg 7822 ltmprr 7845 recexgt0sr 7976 mulgt0sr 7981 map2psrprg 8008

W3C validator